Pretty much what the laconic says. It can be used to give Awesome Moments to some of the characters for cleverness, particularly the Smart Guy. It can also challenge fans to guess how the heroes are going to find their way through the puzzle and like how drama helps fans connect with the heroes emotionally, this trope can help them connect mentally as well.

It's often found in Shōnen (Demographic) manga, usually in the form of fights or Serious Business sporting games. But is definitely not limited to those. It's also a given in most adventure and puzzle games so it might be easier not to list those. They even have a subtrope of this. It can also be used for a mystery case.

Examples:

Spiral runs on this trope. From the beginning it's played in investigations, later it's played for Absurdly HighS takes Games to test the The Hero and even later, applied to gun fights. Any episode in the anime where this isn't present can be counted on one hand.

Hunter × Hunter does this quite a lot. Most notable example being when Gon and his friends are cornered by snakes in a cave and the master who controls them is dead and covered in snakes as well. Gon solved it by reaching into the snake master's pocket, getting bitten by a whole lot of snakes in the process, to get an antidote. The odds were in his favor in that most poison users like to carry an antidote to use as a bargaining chip. Yoshihiro Togashi probably popularized this in the 90s Manga industry.

Most Shounen fighting/sport anime tend to have these half of the time when they don't go with the Deus ex Machina victory route. Examples of fighting series that use these are as follows:

YuYu Hakusho, well it is another Yoshihiro Togashi work. Non fighting examples include the Gate of Betrayal and when Yusuke has to Spot the Imposter among his friends. The imposter could copy their target's appearance and memories. The answer was, which one of them was the most likely to walk into a trap for there to be an imposter to begin with.

The Doraemons Special series does this for almost every fight and unlike most examples, hardly ever goes the dramatic power up victory route. There are non fight examples though. Such as when the gang has to survive a tornado and of all the gadgets Doraemon pulls out, it was a tornado making machine which at first they thought would make it worse as they don't need another tornado. Wangdora however used the tornado making machine to create another tornado with the opposite rotation so the two cancel each other out.

Frank Mentzer's D&D Immortals adventure IM1 The Immortal Storm. A couple of the sub-adventures had logic puzzles, including the initial test by the Hierarchs and the PCs' exploration of the plane of music.

Lawrence Schick's S2 White Plume Mountain. In one room the PCs encounter 5 flesh golems, each with a number on its chest (5, 7, 9, 11, 13). The PCs have 60 seconds to figure out which golem doesn't belong with the others. If they choose the right one it will serve them. If they don't, all of the golems will attack them. The right answer is 9, because the other numbers are all prime numbers.

NUMB3RS uses these for quite a lot of the cases, prominently in the second and third season. Usually the FBI would run into a trouble in their investigation and go to Charlie for help. Charlie then explains and solves the problem(which will occasionally resemble a game theory scenario) with an Exposition Diagram. Sometimes it goes beyond just investigations too, like disarming bombs.

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